## Step 1: Define the symbolic representation of the variables
The variables are defined as follows:
- $x_1$ represents the milligrams of potassium
- $x_2$ represents the milligrams of vitamin B7
- $x_3$ represents the milligrams of vitamin B9
- $x_4$ represents the milligrams of magnesium

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $8.47x_1 + 8.07x_2 + 5.49x_3 + 3.6x_4$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $4x_1 \leq 47$
- $5x_2 \leq 47$
- $3x_3 \leq 47$
- $5x_4 \leq 47$
- $5x_2 + 5x_4 \geq 5$
- $3x_3 + 5x_4 \geq 4$
- $4x_1 + 5x_2 + 5x_4 \geq 8$
- $4x_1 + 5x_2 + 3x_3 \geq 8$
- $4x_1 + 5x_2 + 5x_4 \geq 11$
- $4x_1 + 5x_2 + 3x_3 \geq 11$
- $5x_2 + 3x_3 \leq 27$
- $5x_2 + 5x_4 \leq 36$
- $4x_1 + 5x_2 + 5x_4 \leq 26$
- $4x_1 + 3x_3 + 5x_4 \leq 20$
- $5x_2 + 3x_3 + 5x_4 \leq 31$
- $4x_1 + 5x_2 + 3x_3 \leq 28$
- $4x_1 + 5x_2 + 3x_3 + 5x_4 \leq 28$
- $x_1$ is an integer

## 4: Create the Gurobi model and variables
We will use Gurobi to solve this optimization problem.

## 5: Implement the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="x1", vtype=gp.GRB.INTEGER)  # milligrams of potassium
x2 = m.addVar(name="x2")  # milligrams of vitamin B7
x3 = m.addVar(name="x3")  # milligrams of vitamin B9
x4 = m.addVar(name="x4")  # milligrams of magnesium

# Objective function
m.setObjective(8.47 * x1 + 8.07 * x2 + 5.49 * x3 + 3.6 * x4, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(4 * x1 <= 47)
m.addConstr(5 * x2 <= 47)
m.addConstr(3 * x3 <= 47)
m.addConstr(5 * x4 <= 47)
m.addConstr(5 * x2 + 5 * x4 >= 5)
m.addConstr(3 * x3 + 5 * x4 >= 4)
m.addConstr(4 * x1 + 5 * x2 + 5 * x4 >= 8)
m.addConstr(4 * x1 + 5 * x2 + 3 * x3 >= 8)
m.addConstr(4 * x1 + 5 * x2 + 5 * x4 >= 11)
m.addConstr(4 * x1 + 5 * x2 + 3 * x3 >= 11)
m.addConstr(5 * x2 + 3 * x3 <= 27)
m.addConstr(5 * x2 + 5 * x4 <= 36)
m.addConstr(4 * x1 + 5 * x2 + 5 * x4 <= 26)
m.addConstr(4 * x1 + 3 * x3 + 5 * x4 <= 20)
m.addConstr(5 * x2 + 3 * x3 + 5 * x4 <= 31)
m.addConstr(4 * x1 + 5 * x2 + 3 * x3 <= 28)
m.addConstr(4 * x1 + 5 * x2 + 3 * x3 + 5 * x4 <= 28)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Milligrams of potassium: {x1.varValue}")
    print(f"Milligrams of vitamin B7: {x2.varValue}")
    print(f"Milligrams of vitamin B9: {x3.varValue}")
    print(f"Milligrams of magnesium: {x4.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Symbolic representation of the problem
```json
{
    "sym_variables": [
        ["x1", "milligrams of potassium"],
        ["x2", "milligrams of vitamin B7"],
        ["x3", "milligrams of vitamin B9"],
        ["x4", "milligrams of magnesium"]
    ],
    "objective_function": "8.47*x1 + 8.07*x2 + 5.49*x3 + 3.6*x4",
    "constraints": [
        "4*x1 <= 47",
        "5*x2 <= 47",
        "3*x3 <= 47",
        "5*x4 <= 47",
        "5*x2 + 5*x4 >= 5",
        "3*x3 + 5*x4 >= 4",
        "4*x1 + 5*x2 + 5*x4 >= 8",
        "4*x1 + 5*x2 + 3*x3 >= 8",
        "4*x1 + 5*x2 + 5*x4 >= 11",
        "4*x1 + 5*x2 + 3*x3 >= 11",
        "5*x2 + 3*x3 <= 27",
        "5*x2 + 5*x4 <= 36",
        "4*x1 + 5*x2 + 5*x4 <= 26",
        "4*x1 + 3*x3 + 5*x4 <= 20",
        "5*x2 + 3*x3 + 5*x4 <= 31",
        "4*x1 + 5*x2 + 3*x3 <= 28",
        "4*x1 + 5*x2 + 3*x3 + 5*x4 <= 28"
    ]
}
```